Step 1: Word meaning.
"Hoarse" refers to a rough or harsh voice, usually caused by a sore throat or overuse.
Step 2: Spellings check.
(A) Hoerse: Incorrect.
(B) Hæarse: Wrong spelling.
(C) Hairuse: Incorrect.
(D) Hoarse: Correct spelling.
Step 3: Conclusion.
The correct spelling is (D) Hoarse.